In the first church scene, during which the preacher speaks of loving one's enemy, the choir and congregation go into a song with rhythmic clapping. The video and audio of this clapping are quite noticeably mismatched.
Skeeter is shown using liquid paper, which in 1963 was still being sold out of its inventor's house and was not yet widely available. People used hard, round "typewriter erasers" and brushes to correct typing errors.
The check received by Hilly in the mail from Celia Foote has an endorsement line on the back of the check. That line didn't exist on checks prior to 1987 when it was required by the federal government.
The phones shown are rotary dial and the correct styles and colors available at the time but many have an RJ modular connector on the hand set that was not available until the 1970s. The handset cords and at least one of the extension cords are also the newer flat style, at the time the movie was set they would be thicker and round with both cords usually the same color as the phone set.
Near the end of the scene with the commodes. The closest toilet has a modern plastic fill valve. In the 1960's all of these valves were brass or copper. The color would have been of that metal or a shade of green from corrosion.
The gospel "Victory is Mine" used in the movie and on the soundtrack was actually not written in the 1960's. It was composed by Greg Nelson with a copyright date of 1984, Shepherd's Fold Music, River Oaks Music Company. Also, the style the choir used in the movie in singing the song is clearly too contemporary for the 1960's.
(at around 45 mins) When Skeeter drives into downtown Jackson at the beginning of the film there is a Triumph Motorcycle parked in the background. The color scheme of the motorcycle's tank is the same as the factory color scheme from a 1970 Triumph Tiger 650. The movie takes place around 1962-1963.
When Yule Mae asks the Holbrooks for the advance in wages, Mr. Holbrook quickly excuses himself claiming he's late. As he bends down to give his wife a peck on the cheek, his tie is not clipped to his shirt. After the kiss, and a director's cut, it is properly fixed.
At the fund raiser, Celia rips some of Miss Hilly's dress off. Later when Miss Hilly is outside with her mother, it appears to be fixed again. However, upon closer inspection, the sleeve is clearly pushed up against the rest of the dress to make the damage less conspicuous - a gap can be seen where her skin is showing through.